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 1 

Proceedings of the 2008 IEEE/ASME Joint Rail Conference 

JRC2008 

April 22-23, 2008 Wilmington, Delaware, USA 

 JRC2008-63014 

ANALYSIS OF RAILROAD TANK CAR SHELL IMPACTS USING FINITE ELEMENT 

METHOD 

Y.H. Tang 

Volpe National Transportation Systems Center 

Cambridge, Massachusetts, USA 

H. Yu 

Chenga Advanced Solutions & Engineering, LLC 

Cambridge, Massachusetts, USA 

J.E. Gordon 

Volpe National Transportation Systems Center 

Cambridge, Massachusetts, USA 

D.Y. Jeong 

Volpe National Transportation Systems Center 

Cambridge, Massachusetts, USA 

A.B. Perlman 

Volpe National Transportation Systems Center 

Cambridge, Massachusetts, USA 

ABSTRACT 

This paper examines impacts to the side of railroad tank 

cars by a ram car with a rigid indenter using dynamic, 

nonlinear finite element analysis (FEA). Such impacts are 

referred to as shell impacts. Here, nonlinear means elasticplastic material behavior with large deformations. Several 

computational issues are addressed. The dynamic response of 

the shell structure coupled with the sloshing response of fluid 

inside the tank is characterized through various mesh 

formulations. Puncture of the tank is calculated using a 

material failure criterion based on the general state of stress in 

the shell structure in terms of stress triaxiality. The FEA 

models were verified and validated in previous work. In the 

present work, the verified and validated FEA framework is 

applied to examine the effect of various factors on the 

structural response of the tank. These factors include shell 

thickness and indenter geometry. 

INTRODUCTION 

In its role to provide technical support to the Federal 

Railroad Administration (FRA), the Volpe National 

Transportation Systems Center (Volpe Center) is conducting 

and managing research to examine the structural integrity of 

railroad tank cars under a wide range of conditions that vary 

from the normal operating environment to rare and extreme 

circumstances such as impact loading during accidents [1]. 

A review of accident statistics indicates that the railroad 

tank car industry’s safety performance has generally improved 

over the last forty years. But three recent accidents involving 

the release of hazardous materials have focused attention on 

crashworthiness of railroad tank cars under accident loading 

conditions: (1) a train derailment near Minot, North Dakota 

on January 18, 2002 [2]; (2) a train-to-train collision in 

Macdona, Texas on June 28, 2004 [3]; and (3) a train-to-train 

collision in Graniteville, South Carolina on January 6, 2005 

[4]. 

Further evaluation of accident data indicates that releases 

of toxic inhalation hazard (TIH) materials are attributed to 

failures in three general locations: (1) tank car head, (2) tank 

car shell, and (3) valves and fittings [5]. Between 1965 and 

2005, a total of 252 tank cars released TIH materials in 176 

accidents. Figure 1 shows the distribution of these releases in 

terms of causes and quantity of released commodity. While 

less than half of all releases are caused by failures in the head 

and the shell of pressurized tank cars, such failures account 

for 85 percent of the total gallons of lost lading. Failures to 

valves and fittings account for about one-third of all accidentcaused TIH releases, but less than 5 percent of the total 

gallons of lost lading. 
 

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 2 

Cause of Lading Loss Gallons Lost

Head

23%

Shell

23%

Valves &

Fittings

33%

Other

21%

Head

43%

Shell

42%

Valves & Fittings

4%

Other

11%

Figure 1: Accident-Caused Releases in TIH Tank Cars, 

1965-2005 [5] 

Dynamic, nonlinear (i.e., elastic-plastic material behavior 

with large deformations) finite element analysis (FEA) models 

have been developed to examine the structural response of 

tank cars under generalized head [6] and shell [7] impact 

scenarios. Specifically, the FEA models are used to calculate 

impact force as a function of indentation, or force-indentation 

characteristic. These models were developed using 

commercial finite element codes ABAQUS [8] and LS-DYNA 

[9]. Moreover, the FEA models for shell impact were verified 

and validated in previous work [7]. Verification was 

conducted by comparing results from both solvers with each 

other and with known solutions for static loading. Validation 

was accomplished through comparisons with data obtained 

from full-scale tests that were performed at the Transportation 

Technology Center in Pueblo, Colorado.1

In this paper, the verified and validated FEA models are 

exercised to examine the effect of various factors on the forcedeformation behavior of the tank structure to shell impacts. 

These factors include tank geometry (i.e., shell thickness and 

tank diameter) and indenter geometry. 

DESCRIPTION OF FINITE ELEMENT MODEL 

The FEA models developed to examine tank car impacts 

account for the following physical characteristics of the 

problem: (1) structural dynamics, (2) elastic-plastic material 

behavior with large deformations, (3) fluid-structure 

interaction, and (4) material failure. 

FEA is well established as a computational tool to model 

structural dynamics to simulate how a structure moves with 

time under prescribed loads. Moreover, both ABAQUS and 

LS-DYNA include constitutive models for elastic-plastic 

stress-strain behavior with large deformations. 

When the tank contains lading, the fluid and the tank 

structure both move and exert forces upon one another during 

the impact event. Different mesh formulations are used in the 

 

1

 In the present context, verification refers to the process in ensuring that the 

mathematics are being modeled correctly. Validation refers to the process in 

determining whether physics are being modeled correctly [10]. 

FEA models to account for fluid-structure interaction. 

Specifically, Lagrangian and Eulerian mesh formulations are 

used. An Eulerian mesh is fixed in space, and tracks the 

material passing through. This formulation is used in the LSDYNA impact simulations reported here. The corresponding 

ABAQUS simulations employ a Lagrangian fluid mesh in 

which the fluid nodes follow the material as it deforms. In 

both cases a Lagrangian mesh is used to model the tank 

structure. 

Material failure is predicted in the ABAQUS FEA models 

using a strain-based criterion, referred to as the BaoWierzbicki (B-W) criterion [11]. Figure 2 shows a schematic 

of this failure criterion, which illustrates the assumption that 

plastic strain to initiate failure, εi depends on stress triaxiality,2

η. Failure initiation occurs when loading conditions induce 

effective plastic strains at levels of stress triaxiality above the 

limits suggested by Figure 2. Once failure initiates, damage is 

assumed to propagate in the form of linear strain softening. 

Figure 3 illustrates this concept in which the stress-strain 

behavior of a material element exhibits a linear decrease in 

stress with increasing strain beyond εi. Modeling the 

progression of failure by strain softening helps minimize the 

mesh dependency of the numerical results [12]. The B-W 

failure initiation envelope is shown to consist of three regions 

that represent different modes of failure. Region I is 

associated with high levels of stress triaxiality which promotes 

nucleation, growth and coalescence of voids leading to ductile 

fracture. Region III consists of negative values of stress 

triaxiality which represent shear fracture due to shear band 

localization. Region II comprises positive but low levels of 

stress triaxiality representing mixed mode fracture. The 

schematic also shows that zero stress triaxiality (i.e., η equal 

to 0) is equal to a stress state of pure shear, and that the cusp 

between Regions I and II (which corresponds to η equal to 1⁄3) 

is equal to a stress state of uniaxial tension. In theory, the 

failure initiation envelope for a given material is developed 

through a series of physical tests. The complete series entails 

eleven tests with different specimen geometries to characterize 

different levels of near-constant stress triaxiality in the vicinity 

of failure. Such tests were conducted previously to develop 

failure initiation envelopes for 2024-T351 aluminum [13] and 

A710 steel [14]. In the present implementation, the failure 

initiation envelope for TC-128B tank car steel is constructed 

from a calibration method that uses measurements from 

standard uniaxial tensile tests [15]. According to Lee and 

Wierzbicki [15], this calibration method estimates a failure 

initiation envelope that is within 10 percent agreement of that 

based on the complete test series. 

 

2

 Mathematically, stress triaxiality is the ratio of mean stress to the effective 

or von Mises equivalent stress. Physically, stress triaxiality describes the 

general state of stress. 
 

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Figure 2: Schematic of Fracture Initiation Envelope 

Based on Stress Triaxiality 

Figure 3: Schematic of Linear Strain Softening 

The implementation of the B-W criterion was validated 

using data from pendulum impact tests conducted on 

unnotched Charpy specimens made from TC-128B tank car 

steel [16]. Specimen thickness ranged from 0.19 to 0.83 inch. 

The experimental and simulation results were compared on 

the basis of the energy required to fracture the specimens. 

Results from the FEA simulations were in excellent agreement 

with the experimental observations for the full range of test 

specimen thicknesses. Moreover, the analysis produced 

results within the narrow scatter band of test results for all 

specimens of thickness greater than 0.5 inch. This is 

noteworthy because the range of specimen thickness for which 

the analysis most accurately reproduced the experiment 

corresponds to that used in typical tank car shell construction. 

This favorable outcome suggests that the failure criterion can 

be applied to other cases, such as shell impacts. In modeling 

material failure, accurate results are obtained when solid 

elements are used at and around the impact location. For the 

simulations of pendulum impact tests, sensitivity studies 

suggest that these elements must have an aspect ratio equal to 

one with a minimum of six elements through the thickness. 

This is accomplished by meshing a patch of solid elements of 

sufficient size at the impact location. This patch can be 

coupled to shell elements at locations away from the impact 

point that do not require this special treatment. Shell elements 

in the impact zone do not produce accurate results when used 

in conjunction with the B-W criterion. Applicability of the BW criterion for modeling material failure under loading 

conditions other than those described here is not implied. 

The FEA models for shell impacts do not include tank car 

components such as the manway, body bolsters, and draft sills. 

In addition, the outer steel jacket and thermal insulation are 

neglected. Although the FEA models are simplified with these 

assumptions, exercising the models requires long execution 

times in order to account for structural dynamics, nonlinear 

constitutive material behavior, fluid-structure interaction, and 

material failure. 

COMPARISONS WITH FULL-SCALE IMPACT TESTS 

A series of full-scale tests were conducted to examine the 

structural behavior of tank cars under shell impacts.3

 In these 

tests, a stationary tank car was positioned next to a concrete 

wall. The tank car contained water mixed with clay slurry to 

produce the density approximately equal to that of liquid 

chlorine. The outage in the tank cars used in these tests was 

10.6 percent with an internal pressure of 100 psi. A ram car 

weighing 286,000 lb with a rigid indenter was used to strike 

the side of the tank car. Moreover, these tests were conducted 

to provide data for validating FEA models developed for shell 

impacts. 

The test series comprised three full-scale tests. The first 

test, however, is referred to as an assurance test (also called 

Test 0) because the cars were equipped with limited 

instrumentation. Moreover, Test 0 was conducted to 

understand the test environment and the gross motions of the 

cars. In subsequent full-scale tests, the test cars were heavily 

instrumented to provide redundant measurements for forces 

and displacements, which could be used to validate the FEA 

models. 

Test 0 and Test 1 used an indenter with a rectangular face; 

17 inches in height by 23 inches in width and 1-inch radius on 

each edge. The impact velocity was 10 mph in Test 0, and 14 

mph in Test 1. Moreover, tank integrity was maintained in 

both tests (i.e., no puncture). In Test 2, however, a smaller 

indenter was used (6 inches by 6 inches with 1⁄2-inch edge 

 

3

 The full-scale shell impact tests were conducted in support of an industry 

research-and-development effort called the Next-Generation Rail Tank Car 

Project. Dow Chemical Company, Union Pacific Railroad, and Union Tank 

Car Company are the industry sponsors of this collaboration. The FRA and 

the Volpe Center participate in this project through a Memorandum of 

Cooperation. 
 

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 4 

radii) to increase the likelihood of puncture. The impact 

velocity in Test 2 was 15 mph, which resulted in puncturing 

the tank. 

Figure 4 compares the force-time history measured in one 

of the full-scale tests with those calculated with the FEA 

models assuming different fluid formulations to account for 

fluid-structure interaction. In previous work [7], the Eulerian 

fluid formulation was shown to provide a more accurate 

representation of the force-time history than the Lagrangian 

formulation. The FEA calculated force lags the measured data 

after about 0.12 seconds. Consequently, the fluid-structure 

interaction capabilities of the commercial software are being 

modified to improve the Lagrangian formulation as well as to 

develop a coupled Eulerian-Lagrangian approach. 

Time (seconds)

0.0 0.1 0.2 0.3 0.4 0.5

Force (kips)

0

200

400

600

800

1000

1200

1400

1600

Processed Data

LS-DYNA, Eulerian

ABAQUS, Lagrangian

Figure 4: Comparison of Force-Time Histories for Test 1 

A key output of the FEA models is the force-indentation 

characteristic which describes the structural behavior of the 

tank as it is struck by an impacting object. Moreover, the area 

under the force-indentation curve represents the energy to 

failure which in turn is related to impact velocity to cause 

failure. The specific point at which failure is expected to 

occur on the force-indentation characteristic depends on 

several factors, such as size and shape of the impacting object 

and mechanical properties of the tank material. 

Figure 5 overlays FEA calculated force-indentation 

curves with those measured in Tests 1 and 2. The agreement 

between the experimental and simulation results is excellent. 

The FEA for Test 2 includes material failure, and was 

performed prior to the test. The drop-off in force after about 

16 inches of indentation characterizes puncturing of the tank. 

Prediction of puncture by FEA was validated by the full-scale 

test. This favorable outcome also provides confidence in 

applying the assumed criterion for material failure and in 

modeling of damage progression. The figure also shows a 

regression curve that was generated from the calculated and 

measured curves for both tests. Moreover, the figure suggests 

that indenter size has a relatively weak effect on forcedeformation behavior for indentations less than 20 inches, but 

indenter size appears to have a relatively strong effect on tank 

car puncture. 

Indentation (inches)

0 5 10 15 20 25 30

Force (kips)

0

200

400

600

800

1000

1200

1400

1600

1800

TEST 1 MEASUREMENT (14.0 mph, LARGE PUNCH)

TEST 1 ANALYSIS (14.0 mph, LARGE PUNCH)

TEST 2 MEASUREMENT (15.1 mph, SMALL PUNCH)

TEST 2 ANALYSIS (15.1 mph, SMALL PUNCH)

Regression Curve

Figure 5: Force-Indentation Curves for Tests 1 and 2 

FEA CASE STUDIES 

The verified and validated FEA framework is applied to 

examine the effect of various factors on the structural response 

of tank cars under shell impacts. Specifically, these factors 

are: impact speed, tank shell thickness, and indenter geometry. 

The impact scenario examined in these case studies is 

identical to the full-scale shell impact test. That is, a 286-kip 

ram car is assumed to impact the side of a stationary 263-kip 

tank car containing fluid with the approximate density of 

liquid chlorine. Outage of 10.6 percent and an internal 

pressure of 100 psi internal pressure are also assumed. 

The ram car is assumed to travel at impact speeds varying 

between 10 and 15 miles per hour (mph). Tank geometry is 

characterized by shell thickness and tank diameter. Three 

shell thicknesses are assumed: 0.500 inch, 0.777 inch, and 

0.975 inch. The inner diameter of the tank is 100.625 inches 

in all cases. The size and shape of the rigid indenter for shell 

impacts is varied. Table 1 lists the dimensions of the three 

indenters assumed in this paper. 

Table 1: Dimensions of Different Indenters 

Indenter Height, H

(inches) 

Width, W

(inches) 

Edge Radius 

(inch) 

1 6 6 1⁄2 

2 12 12 1 

3 17 23 1 
 

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Yield strength, ultimate tensile strength, and percent 

elongation are generally assumed to be equal to the minimum 

requirements for TC-128B tank car steel, which are: 50 ksi 

for yield strength, 81 ksi for ultimate tensile strength, and 20 

percent elongation. Temperature and strain rate effects on 

material stress-strain behavior are neglected. 

Maximum Impact Force

In this section, LS-DYNA FEA results are presented in 

terms of maximum force as a function of impact velocity and 

shell thickness for the different indenters. Figure 6 shows 

FEA results for the 6” by 6” indenter, which are denoted by 

the various symbols. The figure also shows regression curves 

based on a best-fit analysis of the FEA results, which are 

denoted by the various dashed lines. Specifically, the FEA 

results are fitted to a regression formula with the following 

functional form 

max 1

p

o F C= ⋅ v (1) 

where Fmax is the maximum impact force in kips and vo is the 

impact velocity in miles per hour (mph). In addition, p is a 

constant and C1 is assumed to depend on shell thickness and 

indenter. In the present work, p is assumed to be equal to 2⁄3. 

The rationalization for this assumption is based on the physics 

of impact, and is described in the Appendix. 

Similarly, Figure 7 shows FEA results and the 

corresponding regression curves for the 12” by 12” indenter, 

and Figure 8 for the 17” by 23” indenter. As the indenter size 

increases, the peak impact forces increase, but the difference 

in these forces between the smallest (i.e., 6” by 6”) indenter 

and the largest (i.e., 17” by 23”) indenter is at most only 5 

percent. As the shell thickness increases from 0.500 to 0.777 

inch, maximum forces increase by 8 to 13 percent. 

Thickening the shell from 0.777 to 0.975 inch raises the peak 

forces by 6 to 11 percent. As the impact velocity increases 

from 10 to 15 mph, the maximum forces increase by 36 to 43 

percent. Therefore, indenter size has a relatively weak effect 

on the maximum impact force compared to shell thickness and 

impact velocity. 

Maximum Indentation

Results from LS-DYNA FEA case studies are presented 

in this section in terms of maximum indentation as a function 

of impact velocity and shell thickness for the different 

indenters. The following regression formula is used to 

provide a best-fit curve to the FEA results 

2

max 2

p C o δ = ⋅ v (2) 

Impact Velocity (mph)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maximum Force (kips)

0

200

400

600

800

1000

1200

1400

1600

FEA, h=0.500 inch

FEA, h=0.777 inch

FEA, h=0.975 inch

Regression, h=0.500 inch

Regression, h=0.777 inch

Regression, h=0.975 inch

Figure 6: Maximum Force as a Function of Impact 

Velocity and Shell Thickness (6” by 6” Indenter) 

Impact Velocity (mph)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maximum Force (kips)

0

200

400

600

800

1000

1200

1400

1600

FEA, h=0.500 inch

FEA, h=0.777 inch

FEA, h=0.975 inch

Regression, h=0.500 inch

Regression, h=0.777 inch

Regression, h=0.975 inch

Figure 7: Maximum Force as a Function of Impact 

Velocity and Shell Thickness (12” by 12” Indenter) 

Impact Velocity (mph)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maximum Force (kips)

0

200

400

600

800

1000

1200

1400

1600

FEA, h=0.500 inch

FEA, h=0.777 inch

FEA, h=0.975 inch

Regression, h=0.500 inch

Regression, h=0.777 inch

Regression, h=0.975 inch

Figure 8: Maximum Force as a Function of Impact 

Velocity and Shell Thickness (17” by 23” Indenter) 
 

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 6 

where δmax is the maximum indentation in inches and vo is the 

impact velocity in mph. In addition, C2 depends on shell 

thickness and indenter. The exponent p is the same constant 

used in equation (1), which is assumed to be equal to 2⁄3 (see 

Appendix). 

Figure 9 shows the FEA results and the corresponding 

regression curves for the 6” by 6” indenter. Similarly, Figure 

10 shows the results for the 12” by 12” indenter, and Figure 

11 for the 17” by 23” indenter. 

For the same shell thickness and constant impact velocity, 

the maximum indentation decreases as the indenter becomes 

larger, but this difference varies between 9 and 12 percent. 

Thickening the shell from 0.500 to 0.777 inch decreases the 

maximum indentation by 12 to 16 percent. Increasing the 

shell thickness from 0.777 to 0.975 inch reduces the maximum 

indentation by 9 to 12 percent. Conversely, raising the impact 

velocity from 10 to 15 mph produces an increase in the 

maximum indentation by 54 to 63 percent. These results 

indicate that indenter size has a relatively weak effect on the 

maximum indentation compared to shell thickness and impact 

velocity. 

Analysis with Material Failure

The finite element analyses were applied in a limited 

number of cases to calculate the velocity at which puncture is 

expected to occur in shell impacts. The material failure 

criterion developed by Bao and Wierzbicki [11] was used in 

conjunction with ABAQUS FEA models in these cases. 

For the case where the shell thickness is equal to 0.777 

inches, the impact velocity at which puncture is expected to 

occur (i.e., puncture velocity) is estimated to be about 11.9 

mph for the 6” by 6” indenter. In the FEA models with 

material failure, the impacted region of the tank is modeled 

using six solid elements through the shell thickness. The next 

three figures show the evolution of stress triaxiality at three 

different locations on the impact footprint for each of the 

elements through the shell thickness as the indenter penetrates 

and ultimately punctures the tank. Element 1 is on the side of 

the tank shell that is impacted by the indenter, element 6 is 

inside the tank. Figure 12 shows the triaxiality map for the 

lower corner of the impact footprint; Figure 13 for the side 

and Figure 14 for the top. The Bao-Wierzbicki failure 

initiation envelope is overlayed in each of the figures, which 

is represented by the solid red curve without symbols. The 

symbols in each of the figures represent the effective plastic 

strain to initiate failure and corresponding value of stress 

triaxiality at different instants of time. Since the FEA model is 

a half-symmetric model, the stresses at the opposite lower 

corner and the opposite side of the impact zone are equal. 

Moreover, the following observations can be made from 

gleaning the FEA results to generate the triaxiality maps: 

Impact Velocity (mph)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maximum Indentation (inches)

0

10

20

30

40

50

60

FEA, h=0.500 inch

FEA, h=0.777 inch

FEA, h=0.975 inch

Regression, h=0.500 inch

Regression, h=0.777 inch

Regression, h=0.975 inch

Figure 9: Maximum Indentation as a Function of Impact 

Velocity and Shell Thickness (6” by 6” Indenter) 

Impact Velocity (mph)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maximum Indentation (inches)

0

10

20

30

40

50

60

FEA, h=0.500 inch

FEA, h=0.777 inch

FEA, h=0.975 inch

Regression, h=0.500 inch

Regression, h=0.777 inch

Regression, h=0.975 inch

Figure 10: Maximum Indentation as a Function of Impact 

Velocity and Shell Thickness (12” by 12” Indenter) 

Impact Velocity (mph)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maximum Indentation (inches)

0

10

20

30

40

50

60

FEA, h=0.500 inch

FEA, h=0.777 inch

FEA, h=0.975 inch

Regression, h=0.500 inch

Regression, h=0.777 inch

Regression, h=0.975 inch

Figure 11: Maximum Indentation as a Function of Impact 

Velocity and Shell Thickness (17” by 23” Indenter) 
 

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(1) For effective plastic strains below the fracture initiation 

limits, stress triaxiality can vary between negative and 

positive values, roughly between ±1. Physically, this 

means that the progression of damage that leads to failure 

is driven by a combination of shear and normal stresses. 

(2) Each triaxiality map is a snapshot of effective plastic 

strain as a function of stress triaxiality in each element in 

the impact zone at the same instant of time. Therefore, 

failure initiates at the lower corner, progresses along the 

side, and eventually reaches the top of the impact 

footprint. 

 (3) Damage initiates on the inside of the tank (i.e., elements 

farthest away from the indenter) before damage initiates 

on the impacted surface. Moreover, damage to the inside 

of the tank or free surface is created by shear stresses 

while damage to the outside of the tank or the impact 

surface is primarily due to normal stresses. 

 (4) The evolution of stress triaxiality in the FEA simulation 

of the full-scale shell impact tests is similar to that 

produced by FEA of unnotched Charpy specimens under 

pendulum impact loading [16]. 

Two other shell impact cases were examined using FEA 

with material failure. In one case, a puncture velocity of 21 

mph was estimated for the 12” by 12” indenter and shell 

thickness of 0.777 inch. In the other case, a puncture velocity 

of 13 mph was estimated for the 6” by 6” indenter and shell 

thickness of 0.975 inch. 

ESTIMATION OF TANK CAR SHELL PUNCTURE 

A semi-analytical method is developed to estimate 

puncture velocity based on the limited number of ABAQUS 

FEA results that included material failure. In this approximate 

method, puncture is assumed to occur when the shear stress 

exceeds a specified value, which is equivalent to the 

maximum shear stress criterion. A failure criterion based on 

maximum shear stress was used in semi-empirical analysis of 

tank car head puncture [17]. Moreover, the semi-empirical 

approach for tank car head puncture was shown to provide 

reasonable estimates for puncture velocity [18].4

Figure 15 compares the B-W and the maximum shear 

stress criteria in terms of stress triaxiality [19]. The figure 

shows that the effective plastic strains to initiate fracture are 

generally lower for the maximum shear stress criterion than 

the B-W criterion over the range of values for stress triaxiality 

shown in the triaxiality maps. Consequently, puncture 

calculations based on the maximum shear stress criterion are 

expected to be more conservative (i.e., pessimistic) than those 

using the B-W criterion. 

 

4

 The term semi-empirical was used to characterize previous work on tank car 

head impacts in which regression formulas were applied to full-scale test data. 

In the present work, the term semi-analytical is used to characterize the 

application of regression formulas to FEA results that were validated with fullscale test data. 

Stress Triaxiality

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Effective Strain to Fracture Initiation 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1

2

3

4

5

6

h

vo

1

2

3

4

5

6

h

vo

Figure 12: Triaxiality Map for Lower Corner of Impact 

Footprint 

Stress Triaxiality

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Effective Strain to Fracture Initiation 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1

2

3

4

5

6

h

vo

1

2

3

4

5

6

h

vo

Figure 13: Triaxiality Map for Side of Impact Footprint 

Stress Triaxiality

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Effective Strain to Fracture Initiation 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1

2

3

4

5

6

h

vo

1

2

3

4

5

6

h

vo

Figure 14: Triaxiality Map for Top of Impact Footprint 
 

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 8 

General quantitative descriptors of indenter size are the 

area and the perimeter of the contact face. Area is a 

convenient metric to relate impact force to contact stress. 

Perimeter may be more appropriate for the shear mode type of 

failure, and has been used in previous studies to estimate 

puncture forces [17]. Moreover, this latter descriptor of 

indenter size is used in the present semi-analytical approach. 

Therefore, the force at which puncture is expected to occur is 

estimated as 

p u 3 ( 2) ( )b a F CH W r = +− σ h (3) 

where Fp is the puncture force in kips, σu is the ultimate 

tensile strength in ksi, h is the shell thickness in inch, H is the 

height of the indenter face in inches, W is the width of the 

indenter face in inches, and r is the edge radius in inch. In 

addition C3, a, and b are constants that are determined from 

the FEA results with material failure, and are equal to 16.5, 

0.54 and 0.67 respectively. In these FEA results, the ultimate 

tensile strength for TC-128B is assumed to be 85 ksi. In 

applying the semi-analytical approach developed here, the 

puncture estimates assume an ultimate tensile strength 

corresponding to the minimum requirement for TC-128B, or 

81 ksi. Moreover, puncture force is estimated from applying 

equation (3). Puncture velocity is then estimated using the 

calculated puncture force and applying equation (1). For a 

tank car with minimum ultimate tensile strength and shell 

thickness of 0.777 inch impacted by a 6” by 6” indenter, the 

puncture velocity is 11.3 mph, compared to 11.9 mph 

calculated from FEA assuming 85 ksi. 

Figure 16 shows estimated puncture velocities for varying 

shell thicknesses and the three different indenters. This figure 

shows that thickening the shell from 0.500 to 0.777 inch 

increases the puncture velocity by 66 to 69 percent, depending 

on the indenter. In addition, thickening the shell from 0.777 

to 0.975 inch improves the puncture velocity by 22 to 27 

percent. Moreover, indenter size is shown to have a 

significant effect on puncture velocity. The difference in 

puncture velocity between the smallest and largest indenters 

considered in this paper is a factor of 2.5. 

Figure 17 shows results from applying the semi-analytical 

approach to estimate the energy to puncture. The energy to 

puncture is calculated from kinetic energy using the puncture 

velocity and the mass of a moving 286,000-lb ram car. Effects 

of shell thickness and indenter size on energy to failure are 

magnified since energy is proportional to velocity squared. 

For a tank car with minimum ultimate tensile strength and 

shell thickness of 0.777 inch impacted by a 6” by 6” indenter, 

the energy to puncture is 1.2 million ft-lb. 

Stress Triaxiality

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Effective Strain to Fracture Initiation 0.0

0.5

1.0

1.5

2.0

Bao-Wierzbicki Criterion

Maximum Shear Stress Criterion

Figure 15: Comparison of Failure Criteria in Terms of 

Stress Triaxiality 

Shell Thickness (inch)

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Puncture Velocity (mph)

5

10

15

20

25

30

35

6" by 6"

12" by 12"

17" by 23"

Figure 16: Estimated Puncture Velocities for Varying 

Shell Thicknesses and Indenters 

Shell Thickness (inch)

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Energy to Puncture (Million ft-lb)

0

2

4

6

8

10

12

6" by 6"

12" by 12"

17" by 23"

Figure 17: Estimated Energy to Puncture for Varying 

Shell Thicknesses and Indenters 
 

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for 

public release; distribution is unlimited. 

 9 

If the ultimate tensile strength is assumed to be 89 ksi (or 

10 percent higher than the minimum requirement), the 

puncture velocities shown in Figure 16 improve by about 15 

percent. The corresponding energies to puncture would 

increase by more than 30 percent. 

The semi-analytical approach for shell impact does not 

explicitly account for fluid-structure interaction. The effect of 

lading is taken into account through the mass of the tank car. 

This approach is expected to provide lower-bound estimates 

for puncture velocity and energy to puncture. 

CONCLUDING REMARKS 

The FEA results presented in this paper suggest that 

indenter geometry has a relatively weak effect on peak impact 

forces and maximum indentations associated with shell (i.e., 

side) impacts compared to shell thickness and impact velocity. 

However, indenter geometry has a significant effect on 

puncture velocity and energy to failure. 

Moreover, the results presented in this paper can be 

applied to evaluate the structural performance of existing tank 

car designs under shell impact loading conditions. 

Future research will be conducted to refine the FEA 

framework for shell impacts. Ultimately, the refined FEA 

framework will be applied to examine head impacts. 

ACKNOWLEDGMENTS 

Federal Railroad Administration, Office of Research & 

Development sponsored the work described in this paper. Ms. 

Claire Orth is the Chief of the Equipment and Operating 

Practices Division. Mr. Francisco Gonzalez is the FRA 

project manager for research on railroad tank cars. Mr. Eloy 

Martinez also provides technical direction to research on 

railroad tank cars. Mr. David Tyrell of the Volpe Center is 

acknowledged for his contributions to research on improved 

tank car designs. Technical discussions with Dr. Oscar 

Orringer are greatly appreciated. Finally, the authors would 

like to acknowledge Professor Christopher Barkan of the 

University of Illinois at Urbana-Champaign for his assistance 

in providing the Volpe Center with access to computers at the 

National Center for Supercomputing Applications. 

REFERENCES 

[1] Tyrell, D.C., Jeong, D.Y., Jacobsen, K., Martinez, E., 

2007: “Improved Tank Car Safety Research,”

Proceedings of 2007 ASME Rail Transportation Division 

Fall Technical Conference, RTDF2007-46013. 

[2] National Transportation Safety Board, 2004: “Derailment 

of Canadian Pacific Railway Freight Train 292-16 and 

Subsequent Release of Anhydrous Ammonia Near Minot, 

North Dakota; January 18, 2002,” Railroad Accident 

Report NTSB/RAR-04/01. 

[3] National Transportation Safety Board, 2006: “Collision 

of Union Pacific Railroad Train MHOTU-23 with BNSF 

Railway Company Train MEAP-TUL-126-D with 

Subsequent Derailment and Hazardous Materials Release, 

Macdona, Texas, June 28, 2004,” Railroad Accident 

Report NTSB/RAR-06/03. 

[4] National Transportation Safety Board, 2005: “Collision 

of Norfolk Southern Freight Train 192 with Standing 

Norfolk Southern Local Train P22 with Subsequent 

Hazardous Materials Release at Graniteville, South 

Carolina, January 6, 2005,” Railroad Accident Report 

NTSB/RAR-05/04. 

[5] Treichel, T., 2006: “List of Accident-Caused Releases of 

Toxic Inhalation Hazard (TIH) Materials from Tank Cars, 

1965-2005,” RSI-AAR Railroad Tank Car Safety 

Research and Test Project, RA 06-05. 

[6] Jeong, D.Y., Tang, Y.H., Yu, H., Perlman, A.B., 2006: 

“Engineering Analyses for Railroad Tank Car Head 

Puncture Resistance,” Proceedings of 2006 ASME 

International Mechanical Engineering Congress and 

Exposition, IMECE2006-13212. 

[7] Tang, Y.H., Yu, H., Gordon, J.E., Priante, M., Jeong, D.Y., 

Tyrell, D.C., Perlman, A.B., 2007: “Analysis of FullScale Tank Car Shell Impact Tests,” Proceedings of 2007 

ASME Rail Transportation Division Fall Technical 

Conference, RTDF2007-46010. 

[8] ABAQUS User’s Manual, Hibbit, Karlsson & Sorensen 

Inc., Pawtucket, Rhode Island. 

[9] LS-DYNA3D User’s Manual, Version 940, Livermore 

Software Technology Company, Livermore, California. 

[10] American Society of Mechanical Engineers, 2006: 

“Guide for Verification and Validation in Computational 

Solid Mechanics,” ASME V&V 10-2006, New York, NY. 

[11] Bao, Y., Wierzbicki, T., 2004: “On fracture locus in the 

equivalent strain and stress triaxiality space,” 

International Journal of Mechanical Sciences 46, 81-98. 

[12] Hillerborg, A., Modeer, M., Petersson, P.E., 1976: 

“Analysis of crack formation and crack growth in 

concrete by means of fracture mechanics and finite 

elements,” Cement and Concrete Research 6, 773-782. 

[13] Bao, Y., Wierzbicki, T., 2002: “Determination of Fracture 

Locus for the 2024-T351 Aluminum,” Massachusetts 

Institute of Technology, Impact and Crashworthiness 

Laboratory Report No. 81. 

[14] Bao, Y., Bai, Y., Wierzbicki, T., 2004: “Calibration of 

A710 Steel for Fracture,” Massachusetts Institute of 

Technology, Impact and Crashworthiness Laboratory 

Report No. 135. 

[15] Lee, Y.W., Wierzbicki, T., 2004: “Quick Fracture 

Calibration for Industrial Use,” Massachusetts Institute of 

Technology, Impact and Crashworthiness Laboratory 

Report No. 115. 

[16] Yu, H., Jeong, D.Y., Gordon, J.E., Tang, Y.H., 2007: 

“Analysis of Impact Energy to Fracture Unnotched 

Charpy Specimens Made from Railroad Tank Car Steel,” 
 

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for 

public release; distribution is unlimited. 

 10 

Proceedings of the 2007 ASME Rail Transportation 

Division Fall Technical Conference, RDTF2007-46038. 

[17] Shang, J.C., Everett, J.E., 1972: “Impact Vulnerability of 

Tank Car Heads,” Shock and Vibration Bulletin 42, 197-

210. 

[18] Jeong, D.Y., Tang, Y.H., Perlman, A.B., 2001: 

“Evaluation of Semi-Empirical Analyses for Railroad 

Tank Car Puncture Velocity, Part I: Correlations with 

Experimental Data,” Volpe Center Final Report, 

DOT/FRA/ORD-01/21.1. 

[19] Wierzbicki, T., Bao, Y., Lee, Y.W., Bai, Y., 2005: 

“Calibration and evaluation of seven fracture models,” 

International Journal of Mechanical Sciences 47, 719-

743. 

APPENDIX 

This appendix describes an energy balance to derive 

expressions for maximum force and maximum indentation as 

functions of impact velocity. Moreover, the expressions 

developed in this derivation are intended to show the 

functional form of the equations used to perform best-fit 

regression analysis to the FEA results presented in this paper. 

 In this energy balance, kinetic energy of the moving ram 

car is equated to strain energy due to deformation of the 

stationary tank car. Kinetic energy is defined as 

2

1

1

2

K o E mv = (4) 

where m1 is the mass of the ram car and vo is the impact 

velocity. 

Strain energy is equal to the area under the forceindentation curve, or 

0

max

SE F x dx ( )

δ

= ∫ (5) 

The functional form of the force-indentation curve is assumed 

as 

n F = kδ (6) 

where n is a constant and k depends on shell thickness.5

 

Strain energy is then equal to 

max

1

max

0

1

1

n n SE kx dx k

n

δ

δ += = ∫

+

 (7) 

 

5

 In theory, k also depends on other factors such as tank diameter and internal 

pressure, which were not varied in the present study. 

An expression for maximum indentation as a function of 

impact velocity can be derived by equating kinetic energy, 

equation (4) and strain energy, equation (7) 

1

1 1 2

max

1

2

n

o

n m v

k

δ

+ +

= ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ (8) 

In addition, an expression for maximum force as a function of 

impact velocity can be derived as6

1 1 2

max

1

2

n

n

o

n m Fk v

k

+ +

= ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ (9) 

Moreover, the force-indentation data from the full-scale 

tests and the results from the FEA (refer to Figure 5) suggest 

that n, the exponent in equation (6), is equal to 1⁄2. Applying 

this value to equations (8) and (9) further suggests that the 

maximum indentation is proportional to the impact velocity 

raised to the 4/3 power, and that the maximum force is 

proportional to the impact velocity raised to the 2/3 power. 

 

6

 This derivation implicitly assumes that the maximum force and maximum 

indentation occur simultaneously.